The corresponding asymptotics for the functional \(I\left[\hat{L}_n^{(\alpha )}\right]\)  is given, according to (REF ) and taking into account that \(\psi (z) \sim \log z - \frac{1}{2z}\)  for \(z \rightarrow \infty \)  (see Eq. 5.11.2 of [1]}) and \(\alpha \log (\alpha +n+1) = \,\alpha \log (\alpha )+n+1+\mathcal {O}(\alpha ^{-1})\) , as
\(I\left[\hat{L}_n^{(\alpha )}\right] = \log \left(\frac{e}{\alpha }\right)^{\alpha }+n+\frac{1}{2}+\mathcal {O}(\alpha ^{-1}), \qquad \alpha \rightarrow \infty .\) 
